Q3 of 30 Page 70

4n – 1 is divisible by 3, for each natural number n.

Given; P(n) = 4n – 1 is divisible by 3.

P(0) = 40 – 1 = 0; is divisible by 3.


P(1) = 41 – 1 = 3; is divisible by 3.


P(2) = 42 – 1 = 15; is divisible by 3.


P(3) = 43 – 1 = 63; is divisible by 3.


Let P(k) = 4k – 1 is divisible by 3;


⇒ 4k – 1 = 3x.


⇒ P(k+1) = 4k+1 – 1


= 4(3x + 1) – 1


= 12x + 3; is divisible by 3.


⇒ P(k+1) is true when P(k) is true


∴ By Mathematical Induction P(n) = 4n – 1 is divisible by 3 is true for each natural number n.


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